Properties

Label 16T13
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_{8}$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $13$
Group :  $D_{8}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(13,15)(14,16), (1,4,6,8,10,12,14,15)(2,3,5,7,9,11,13,16)
$|\Aut(F/K)|$:  $16$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $D_{8}$ x 2

Low degree siblings

8T6 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 6, 8,10,12,14,15)( 2, 3, 5, 7, 9,11,13,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$
$ 8, 8 $ $2$ $8$ $( 1, 8,14, 4,10,15, 6,12)( 2, 7,13, 3, 9,16, 5,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 7]
Character table:   
     2  4  2  2  3  3  3  4

       1a 2a 2b 8a 4a 8b 2c
    2P 1a 1a 1a 4a 2c 4a 1a
    3P 1a 2a 2b 8b 4a 8a 2c
    5P 1a 2a 2b 8b 4a 8a 2c
    7P 1a 2a 2b 8a 4a 8b 2c

X.1     1  1  1  1  1  1  1
X.2     1 -1 -1  1  1  1  1
X.3     1 -1  1 -1  1 -1  1
X.4     1  1 -1 -1  1 -1  1
X.5     2  .  .  . -2  .  2
X.6     2  .  .  A  . -A -2
X.7     2  .  . -A  .  A -2

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2